Optimal Polynomial Energy Decay and Spectrum-Determined Exponential Stability for an Euler–Bernoulli Beam With Boundary Memory: Spectral and Sensitivity Analysis

This paper investigates the stability of an Euler–Bernoulli beam subject to boundary memory and highlights the fundamentally different mechanisms associated with polynomially and exponentially decaying resolvent kernels. The formulation distinguishes the original memory kernel, its Volterra resolvent, the compatibility conditions, and the corresponding physical energy. For polynomial memory, a positive original kernel is constructed, and a direct two-step weighted-multiplier method is developed for compatible trajectories issued from zero prehistory. This yields a polynomial energy-decay rate together with a matching lower bound for trajectories with nontrivial tip displacement, proving optimality of the decay exponent within this physical class. A positive diffusive quadrature is also introduced and shown to preserve the semidiscrete dissipation structure. For exponential memory, the history term admits a finite-dimensional realization. Semigroup and Lyapunov arguments establish exponential stability. The spectral analysis combines an analytic characteristic determinant, local Rouché arguments, comparison with a conservative reference beam, and a Riesz basis theorem, leading to spectrum-determined growth and an explicit relation between the uniform energy-decay rate and the spectral abscissa. Finally, Sobol indices, repeated Saltelli estimates, and derivative-based measures quantify the relative influence of the parameters on the high-frequency spectral margin.MSC2020 Classification: 74K10, 74D05, 35B35, 65C05.